To convert radical to exponential form, let us first consider the equivalent form of radical.
Convert the radical form to exponential form :
Problem 1 :
(^{5}√x)^{3}
Solution :
5^{th} root can be written as power 1/5.
(^{5}√x)^{3} = (x^{1/5})^{3}
To simplify the term, which is having a power raised to another power, we can multiply the powers.
(^{5}√x)^{3 }= x^{(1/5) x }^{3}
= x^{3/5}
Problem 2 :
(^{3}√y^{4})
Solution :
Cube root can be written as power 1/3.
(^{3}√y^{4}) = (y^{4})^{1/3}
= y^{4/3}
Problem 3 :
(√x^{5})
Solution :
Square root can be written as power 1/2.
(√x^{5}) = (x^{5})^{1/2}
= x^{5/2}
Problem 4 :
(^{3}√5)^{3}
Solution :
Cube root can be written as power 1/3.
(^{3}√5)^{3} = (5^{(}^{1/3)})^{3}
= 5^{3/3}
= 5
Problem 5 :
√(16x^{2})
Solution :
Square root can be written as power 1/2.
√(16x^{2}) = (16x^{2})^{1/2}
16 can be written as 4^{2}
√(16x^{2}) = (4^{2}x^{2})^{1/2}
= ((4x)^{2})^{1/2}
= (4x)^{2 }^{⋅}^{ (1/2)}
= 4x
Problem 6 :
1/(√(6x))^{3}
Solution :
Square root can be written as power 1/2.
1/(√(6x))^{3 }= 1/[(6x)^{1/2}]^{3}
= 1/(6x)^{(}^{1/2) x }^{3}
= 1/(6x)^{(}^{3/2)}
When we move the term from numerator to denominator or denominator to numerator, we have to change the sign of the power.
= (6x)^{-3/2}
Problem 7 :
1/(^{4}√x)^{7}
Solution :
Fourth root can be written as power 1/4.
1/(^{4}√x)^{7 }= 1/(x^{1/4})^{7}
= 1/x^{(}^{1/4}^{) }^{⋅ }^{7}
= 1/x^{(7}^{/4}^{)}
= x^{-7/4}
Problem 8 :
1/√(5x)
Solution :
Square root can be written as power 1/2.
1/√(5x) = 1/(5x)^{1/2}
= (5x)^{-1/2}
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